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Math Help - Mathematical Induction Question

  1. #1
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    Mathematical Induction Question

    Hello, I have this problem that I can't figure out. I don't think I have too much issue with a simple version of maths induction but this particular question I can't seem to figure out.

    The question is:

    Use mathematical induction to prove that for every positive integer
    n

    1^
    3 + 2^3 + 3^3 + . . . + n^3 = ((n (n + 1))/2)^2

    I can only get to this stage:

    ((n(n + 1))/2)^2 + (n + 1)^3 = (((n + 1)((n + 1) + 1))/2)^2

    I would greatly appreciate it if some wiz could help me out.

    Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by madmunk View Post
    Hello, I have this problem that I can't figure out. I don't think I have too much issue with a simple version of maths induction but this particular question I can't seem to figure out.

    The question is:

    Use mathematical induction to prove that for every positive integer
    n
    1^
    3 + 2^3 + 3^3 + . . . + n^3 = ((n (n + 1))/2)^2

    I can only get to this stage:

    ((n(n + 1))/2)^2 + (n + 1)^3 = (((n + 1)((n + 1) + 1))/2)^2

    I would greatly appreciate it if some wiz could help me out.

    Thanks.
    Expand the left hand side:
    \frac{1}{4} ( n^4 + 2n^3 + n^2 ) + (n^3 + 3n^2 + 3n + 1)

    = \frac{1}{4} ( n^4 + 6n^3 + 13n^2 + 12n + 4 )

    Now the right hand side:
    \frac{1}{4}(n^2 + 2n + 1)(n^2 + 4n + 4) = \frac{1}{4} \left [ (n^4 + 4n^3 + 4n^2) +( 2n^3 + 8n^2 + 8n) + (n^2 + 4n + 4) \right ]

     = \frac{1}{4} ( n^4 + 6n^3 + 13n^2 + 12n + 4 )

    -Dan
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  3. #3
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    Thanks Dan, I understand now.
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